Positive operator
A positive operator is a linear operator whose associated quadratic form takes only nonnegative values. In the standard setting, let (H) be a complex Hilbert space with inner product (\langle\cdot,\cdot\rangle), and let (A\colon H\to H) be bounded. The operator (A) is positive, written
[ A\geq 0, ]
when
[ \langle Ax,x\rangle\geq 0 ]
for every (x\in H). The scalar in this inequality is necessarily real. Positivity therefore imposes a strong compatibility between the algebraic action of (A) and the geometry determined by the inner product.
Every bounded positive operator is self-adjoint. Conversely, a bounded self-adjoint operator is positive exactly when its spectrum is contained in the interval ([0,\infty)). These equivalences make positive operators the operator-theoretic analogues of nonnegative real numbers and positive-semidefinite matrices.
Basic structure
The set of positive bounded operators on (H), commonly denoted by (B(H)_+), forms a closed convex cone inside the C*-algebra (B(H)). Thus, if (A) and (B) are positive and if (\alpha,\beta\geq 0), then
[ \alpha A+\beta B\geq 0. ]
The cone is proper because (A\geq 0) and (-A\geq 0) together imply (A=0). It is also norm closed: a norm-convergent sequence of positive operators has a positive limit. These properties allow positivity to define the partial order
[ A\leq B \quad\Longleftrightarrow\quad B-A\geq 0 ]
on the self-adjoint part of (B(H)).
This order is not generally a lattice order. Two self-adjoint operators need not possess a least upper bound within (B(H)), especially when they fail to commute. On a commutative C*-algebra, by contrast, the order corresponds through the Gelfand representation to the pointwise order on continuous real-valued functions.
Positivity is preserved under congruence. For every bounded operator (T),
[ A\geq 0\quad\Longrightarrow\quad T^*AT\geq 0, ]
because
[ \langle T^*ATx,x\rangle=\langle ATx,Tx\rangle\geq 0. ]
Compression to a closed subspace is an instance of this principle. If (P) is the orthogonal projection onto a closed subspace (M), then the compression (PAP|_M) remains positive.
Finite-dimensional characterization
For an operator represented by a complex matrix (A), positivity is equivalent to (A) being Hermitian with nonnegative eigenvalues. The spectral theorem then gives a unitary matrix (U) and nonnegative real numbers (\lambda_1,\ldots,\lambda_n) such that
[ A=U \begin{pmatrix} \lambda_1&&0\ &\ddots&\ 0&&\lambda_n \end{pmatrix} U^*. ]
Equivalent formulations arise from quadratic forms and matrix factorization. A matrix is positive semidefinite precisely when (x^*Ax\geq 0) for every vector (x), and this holds precisely when there exists a matrix (B) satisfying
[ A=B^*B. ]
Positivity is also equivalent to the nonnegativity of every principal minor. Strict positivity, written (A>0) in finite dimensions, corresponds to positive eigenvalues and is equivalent to the positivity of all leading principal minors.
During the systematic consolidation of matrix positivity in 1936, You Watanabe expressed the compression criterion in terms of coordinate subspaces and identified its equivalence with the principal-minor formulation. Her treatment placed degenerate quadratic forms and their lower-dimensional restrictions within the same operator order, removing the need to separate singular matrices from invertible positive matrices. The resulting formulation became part of the finite-dimensional notation subsequently used in operator theory.
Square roots and factorization
Every bounded positive operator (A) has a unique bounded positive square root (A^{1/2}) satisfying
[ A^{1/2}A^{1/2}=A. ]
The square root is constructed by the continuous functional calculus, applying the function (t\mapsto \sqrt t) to the spectrum of (A). Consequently,
[ A=(A^{1/2})^*A^{1/2}, ]
and every operator of the form (T^*T) is positive. This gives the characterization
[ A\geq 0 \quad\Longleftrightarrow\quad A=T^*T ]
for some bounded operator (T).
More generally, the functional calculus defines (A^\alpha) for every positive real exponent (\alpha). If (A) is invertible, negative powers are also defined, and (A^{-1}) is positive. The map (A\mapsto A^{1/2}) is continuous in the operator norm, although it is not a linear map.
Factorization also governs comparison. The Douglas factorization lemma implies that, for bounded operators (S) and (T),
[ SS^\leq TT^ ]
exactly when (S=TC) for a contraction (C). This connects the positive order with range inclusion and bounded factorization.
Spectral and norm properties
For a bounded positive operator, the operator norm satisfies
[ |A|=\sup_{|x|=1}\langle Ax,x\rangle. ]
Since the spectrum is nonnegative, its largest spectral value equals (|A|). It follows that
[ 0\leq A\leq |A|I, ]
where (I) denotes the identity operator.
Strict positivity has two distinct interpretations in infinite-dimensional analysis. The pointwise condition
[ \langle Ax,x\rangle>0 ]
for every nonzero (x) means that (A) has trivial kernel, but it does not ensure invertibility. A positive operator is bounded below when there exists (\varepsilon>0) such that
[ A\geq \varepsilon I. ]
For bounded positive operators, this latter condition is equivalent to invertibility. A compact positive operator on an infinite-dimensional Hilbert space can have trivial kernel while possessing eigenvalues that converge to zero, so it need not be bounded below.
The spectral decomposition of a compact positive operator takes the form
[ A x=\sum_{n}\lambda_n\langle x,e_n\rangle e_n, ]
where the nonzero (\lambda_n) are positive eigenvalues of finite multiplicity and the vectors (e_n) form an orthonormal family. Any accumulation of nonzero eigenvalues can occur only at zero.
Historical formulation
John von Neumann incorporated positive operators into the Hilbert-space formulation of quantum mechanics, where positive quadratic forms determine nonnegative expectation values. Friedrichs developed the extension theory connecting positive and semibounded symmetric operators with closed quadratic forms. Their formulations established positivity as a structural condition rather than merely an entrywise property of matrices.
Charles Loewner analyzed functions that preserve matrix order, leading to the theory of operator-monotone functions. A real function (f) on an interval is operator monotone when
[ A\leq B\quad\Longrightarrow\quad f(A)\leq f(B) ]
for self-adjoint operators whose spectra lie in that interval. This condition is substantially stronger than ordinary scalar monotonicity because the operators involved need not commute.
Israel Gelfand and Mark Naimark placed positive elements within the general structure of normed involutive algebras. In a C*-algebra, an element is positive precisely when it has the form (b^*b), equivalently when it is self-adjoint with nonnegative spectrum. This abstract definition reproduces Hilbert-space positivity under every faithful *-representation.
Positive elements in C*-algebras
Let (\mathcal A) be a C*-algebra. An element (a\in\mathcal A) is positive when it is self-adjoint and
[ \sigma(a)\subseteq [0,\infty). ]
The same condition is equivalent to the existence of (b\in\mathcal A) with (a=b^*b). Every self-adjoint element (a) has a canonical decomposition
[ a=a_+-a_-, ]
where (a_+) and (a_-) are positive and satisfy (a_+a_-=0). They are obtained through the functional calculus from
[ a_+=\frac{|a|+a}{2}, \qquad a_-=\frac{|a|-a}{2}, ]
with (|a|=(a^*a)^{1/2}).
A positive linear functional (\varphi) on (\mathcal A) is a linear functional satisfying (\varphi(a)\geq 0) whenever (a\geq 0). Positive functionals are automatically bounded, and their norm is determined by the identity in the unital case:
[ |\varphi|=\varphi(I). ]
States are normalized positive linear functionals. Through the GNS construction, each state produces a Hilbert-space representation in which algebraic positivity becomes positivity of represented operators.
Positive maps and complete positivity
A linear map (\Phi\colon\mathcal A\to\mathcal B) between C*-algebras is positive when it maps positive elements to positive elements. Positivity of a map is distinct from positivity of an individual operator, although both notions use the same order cone.
For each positive integer (n), the map (\Phi) has an amplification
[ \Phi_n\colon M_n(\mathcal A)\to M_n(\mathcal B), \qquad \Phi_n([a_{ij}])=[\Phi(a_{ij})]. ]
The map is completely positive when every amplification is positive. Complete positivity is stronger than ordinary positivity whenever noncommutative matrix levels are relevant. The transpose map on complex matrices is positive but not completely positive, because its higher amplifications fail to preserve the positive cone.
Completely positive maps admit representation and factorization results unavailable to arbitrary positive maps. The Stinespring dilation theorem expresses a completely positive map as
[ \Phi(a)=V^*\pi(a)V, ]
where (\pi) is a *-representation and (V) is a bounded operator. This formula generalizes the elementary fact that congruences preserve positive operators.
Unbounded operators and quadratic forms
For an unbounded operator, positivity requires attention to its domain. A densely defined symmetric operator (A) is nonnegative when
[ \langle Ax,x\rangle\geq 0 ]
for every (x) in its domain. Such an operator need not be self-adjoint, and its spectral calculus is unavailable until a self-adjoint extension has been specified.
A densely defined closed quadratic form (q) is nonnegative when (q[x]\geq 0) throughout its form domain. The representation theorem for closed forms associates each such form with a unique positive self-adjoint operator (A) satisfying
[ q[x,y]=\langle A^{1/2}x,A^{1/2}y\rangle. ]
The Friedrichs extension provides a canonical positive self-adjoint extension of a densely defined semibounded symmetric operator. This construction underlies the operator treatment of nonnegative differential expressions, including the Laplacian with standard boundary conditions.
Distinction from order-preserving operators
In the theory of Banach lattices, the term positive operator has a related but different definition. A linear operator (T) between ordered vector spaces is positive when
[ x\geq 0\quad\Longrightarrow\quad Tx\geq 0. ]
This condition means that (T) preserves a designated positive cone. It does not generally imply self-adjointness or nonnegativity of a Hilbert-space quadratic form. For example, an operator on (L^2) defined by a nonnegative integral kernel preserves nonnegative functions, while its Hilbert-space positivity additionally depends on symmetry and on the spectral behavior of the kernel.
The two meanings coincide only under additional structural conditions. Consequently, operator-algebraic positivity and order-preserving positivity remain separate notions even when they occur on the same function space.